BALTIC-WAY 2022 Problem 13

Let $ABCD$ be a cyclic quadrilateral with $AB < BC$ and $AD < DC$ . Let $E$ and $F$ be points on the sides $BC$ and $CD$ , respectively, such that $AB = BE$ and $AD = DF$ . Let further M denote the midpoint of the segment $EF$ . Prove that $\angle BMD = 90^o$ .